JULY 1, 1989 PHYSICAL REVIEW
DIATOMIC MOLECULES ACCORDING TO THE WAVE
MECHANICS. II. VIBRATIONAL LEVELS
BY PHILIP M. MORSE
Palmer Physical Laboratory, Princeton University
(Received April 8, 1929)
ABSTRACT
An exact solution is obtained for the Schroedinger equation representing the
motions of the nuclei in a diatomic molecule, when the potential energy function
is assumed to be of a form similar to those required by Heitler and London and
others. The allowed vibrational energy levels are found to be given by the f'orrnula
E(n) =E,+hcoo(n+1/2) —kcoox(n+1/2)', which is known to express the experimental
values quite accurately. The empirical law relating the normal molecular separation
ro and the classical vibration frequency oro is shown to be ro coo =X to within a probable
error of 4 percent, where Z is the same constant for all diatomic molecules and
for all electronic levels, By means of this law, and by means of the solution above, the
experimental data for many of the electronic levels of various molecules are analyzed
and a table of constants is obtained from which the potential energy curves can be
plotted. The changes in the above mentioned vibrational levels due to molecular
rotation are found to agree with the Kratzer formula to the erst approximation.
INTRoDUcTIQN
'HE wave equation for the nuclear motion' of a diatomic mocluele of
nuclear masses 3f& and M2 and charges Z& and Z&, is approxim;ately
Svr'p
p'P+ —[W —(e'Z~Z, /r)+ V,(r) ]P=0
h'
where p= M~3Ie/(M&+&2), r is the distance between the nuclei, 0 and @
the usual orientation angles measured from the center of gravity, and U, (r)
the electronic energy calculated by considering the two nuclei as fixed in
space a distance r apart.
The combination of the energy of repulsion and the electronic energy
can be considered as a nuclear potential energy
E(r) =(e'Z&Z2/r) V,(r) .—
The wave function 0' can be considered as a product of three factors
O'= N C (P) 0'(0) R(r)/r, where it can be shown that
4 =e"&
O~ =sing 8 Pf(cos 0)
where g and j are integers. The normalizing factor X is adjusted so that
J'Ir 4 du=1.
When these functions have been substituted in the general equation an
equation for R results,
' Born and Oppenheimer, Ann. d. Physik 84, 457 (1927).
PIIII IP 3II. AHORSE
d'E jj+I 2 8~'p
+ + [W —Z(r)]a=0. (2)
Since we are primarily interested in vibrational levels, j is set equal to zero.
The function E(r) is a complicated function of r and of the electronic
quantum numbers, and is not known accurately for any molecule, so it is
useless to try to substitute the actual expression for E in the equation. It
is possible, however, to assume some functional form for E which gives
curves of approximately the same shape as the actual curves, and whose
constants can be carried through the calculations to the end and then adjusted
to conform to the experimental data.
The forms for E which have been used are, among others, the two series-"
a/r+b(r'+c(r ro)'+— , and b'(r r,)—'+c' (r ro)'+These
series give a general equation for the allowed energy levels
8'= —D+hcu, [(e+-',) —x(e+ ', )'+E (-0+', )'+ -] (3a)
where the constants coo, x, E, , are functions of a, b, c, etc., and so if
8' is known empirically, E can be determined.
There are several objections to such series forms for E. In the first
place the effect of all the terms in (r-r, ) to the power 3 or over (which terms
are not always small) must be computed by perturbation methods, thus
adding another approximation to a list already long. In the second place
the series for E determined from the known values of ~0, x, E3, etc., does
not converge for large values of r, and so the series is only applicable over a
restricted range of r.
In the third place the experimental data show that constants E3
are very much smaller than coo or x, whereas the general series for E do not
show that any such peculiarities should exist. In other words these series
are too general.
K'e must search, then, 'for a form for E which will satisfy the following
requirements: (I), It should come asymptotically to a finite value as
r~~; (2), It should have its only minimum point at r ro, (3=), It should
become infinity at r =0 (this need not be exactly true, however, the results
are practically the same if E becomes very large at r =0); (4), It should
exactly give the allowed energy levels as the finite polynomial.
W(e) = D+hcoo [(e—+-',) —x(e+-')'] (3)
The very small correction term coefficients E3,can then be determined
by perturbation methods with a reasonable expectation that these methods
on such small quantities will give fairly accurate results.
A form will be chosen for E which satisfies requirements I and 2 exactly,
and 3 approximately, and the problem will then be to show that the chosen
form satisfies requirement 4.
A SQLUTIQN oF THE PRoBLEM
The function which it is proposed to use here is the simple one
g(r) —Dc—2a(r—ro) 2Dc—u(r—ro)
Fues, Ann. d. Physik 80, 367 (1926).
(4)
VIBRATIONAL LEUELS OIi DIATOMIC MOLECULZS
This function has a minimum of —D at r =ro, comes asymptotically to zero
at r = oo and in general gives curves of a very similar form to the few potential
energy curves which have been calculated theoretically. '4' The only
portion where it does not fit these curves is at x=0, where it should be infinity.
But it will be seen that for the values of D and a used to fit the data
R(r) is between 100D and 10000D at r =0, a value so large that, as far as
its effect on the energy levels and wave function goes, it is as good as in-
finity.
The frequency of classical small vibrations about ro is
~0 =(a/2~) (2D// )'"
If this form of E is substituted in Eq. (3), j set equal to zero, and the
transformation u= (r —re)made, then
d2R Sx'p
——+ [Ilr De
—""+2De ~"]R=0
aN' h'
(6)
The boundary conditions are now set that R must be finite, single valued
and continuous in the range —~ & u ~ +~. It will be found that for some
allowed solutions in this case r4' will not be zero at r =0. But in every case
r%' will be extremely small, and since the point r =0 is some distance outside
the important interval where W)E this discrepancy will not affect the
values of the energy levels.
In other words, since we have admittedly not used the correct form for
E, and since we presumably could not find the true solution for R anyway,
we must content ourselves with a solution which deviates from the correct
solution in a portion which has little effect on the values of the allowed
energies, especially since this deviation is very small.
Make a second transformation, letting y =e ". Then
d
O'R 1 dR Sm'p W 2D
+— + ——+ —D R=O (&)
where now R must be finite, continuous and single valued over the range
~ &y~0. Let R=e '" (2dy)"' F(y). Then if
d =2s-(2/dD) "'/ah
W = — h'ab'/32 'ps
and if z =2dy, then the equation becomes
87r'IJ,D
(d'd/d ')+(d+) —)/dd/d )+ — —d/2 —1 2)F/= . 0
a'dh'
(8)
(9)
The solution of this equation is a finite polynomial' if (87r'pD/a'h'd —b/2
—1/2) =n, an integer greater than zero. That is
b =4r/(2/dD) '/'/ah —1 —2n = h —1 —2n
' 0. Burrau, Klg. Danske. Vid. Selskab. 7, 14 (1927).
' Heitler and London, Zeits. f. Physik 44, 455 (1927).
~
Morse and Stueckelberg, Phys. Rev. 33, 907 (1929).
' Schroedinger Ann, d. Physik 80, 483 (1926).
60 PHILIP M'. MORSE
where k (=4&r(2/aD))/'/ah) must be greater than unity to have a discrete
energy spectrum. The positive values of b are the only ones for which R is
finite over the range O~s( ~. This means that n can have any integral
value in the range 0&2n((k —1).
The solutions for Ii are the generalized Laguerre polynf mials
d b dn+b
F=L„+),(s) =—e*. (s"+'e *)
S b dZn+b
Usually both superscript and subscript-minus-superscript are taken to
be integers, although the superscript need not be integral. Here b is a fraction,
and fractional differentiation must be used to obtain the polynomial
from the definition above. The polynomial can be obtained by the use of
the recursion formula obtained directly from Eq. (10) without evoking the
use of fractional differentiation, but it is felt desirable to use the general
definition for the sake of uniformity. Such differentiation does not vitiate
the general formulas of integration etc., which have been developed for
these polynomials, as long as n is an integer.
The formulas needed for fractional differentiation are given here for
convenience.
d'(s~) I'(a+1) d'(e ')
&a
—b ~ —&i'/rb. g
—z
ds' I'(a —b+ I) ds'
The application of this to the definition of F results in
I'(k —e)
L q ( ))=se/x&k —n —1& &n (k n —1)—ex~
n.~
(k —e —1)(k —e —2)e(e —I)x" '
+ [2!
The normalizing integral
f s' 'e 'L„+b(s) L +/, (s) ds=ll/„
'0 (if nWn&)
I'(k —2n+s —1)
[I'(k —e) ]' Q ——(if e=n/) (12)
I'(s —1)
so that the complete wave function R for the nuclear vibration is
(2da/Q )1/2, e- ee ~/" "&&
[2de
—u(~—ro)
](&:—2e—1&/2 IIc—2n —1[2de—a(~ ro) ]-a—n—1
The square of this, times the perturbing energy, is to be multiplied by dr
as a "volume element" and the integral taken from zero to infinity to obtain
the perturbation energy.
The allowed energy levels are obtained from Eqs. (9) and (11)
W(e) = —a'h'(k —1 —2e)'/32m'p
ah
= —D+—(n+ 1/2) (2D/p) '/ —a'h'(n+ 1/2)'/8&r p (13)
2'
D+h/ee(n+ 1—/2) —(h'/ee'/4D) (n+ 1/2) '
VIBRATIONAL LEVELS OF DIATOllIIIC MOLECULES 61
from Eq. (5). As has been noted before, n takes on all integral values from
zero to (fp —1)/2. This is the first case noted of a Schroedinger equation
giving a finite number of discrete energy levels.
Equation (13) is of the form of empirical Eq. (3), and it therefore is of
the form which we set out to obtain. Since this equation expresses the empirical
data so well in most cases, therefore the potential energy E, as given
by Eq. (4), must have the same shape as the real potential energy throughout
the range where Eq. (13) is a valid representation of the actual energy levels.
Thus if the lists of spectroscopically determined molecular constants
give rp in Angstrom units and up and copx in wave-numbers, D is found in
wave-numbers by the equation
D =ppp /4&ppx
and the coefficient a is found by the relation
a = (8pr'cpco px/h) '"=0.2454(Mcupx) '"
(15)
(16)
where M'= M~Mp/(M~+3IIp), 3II~ and cVp being the atomic weights of the two
nuclei in terms of oxygen 16.
With rp known and D and u determined, the potential energy curve corresponding
to the data is given by Eq. (4), where Z is in wave-numbers if
r is in Angstrom units.
AN EMPIRICAL LAW FOR fp
When the available lists of molecular constants were examined it was
found that in many cases cop and copx were known but rp was not known.
Several writersv have made use of a relation rp'~p=constant to obtain the
.- unknown rp's, but deviations from this relation are quit@ large.
To find what law obtained, if any, between rp and cop, 21 cases were
taken from Birge's table' where rp and orp were both known. An equation,
log cop —p log rp= log E was assumed, and the data were subjected to a least
squares analysis. The most probable values of the constants were found to
be p =2.95 and X=2975. If p be taken to be 3 then the equation becomes
rp'~ p
= 3000A'/cm
to within a probable error of +120 and a maximum deviation of 420. This
is about due to the probable error of the recorded values of rp, for if these
values 1. ad a probable error of 1.3 percent, then rp' would be given to a probable
error of 4 percent.
' Birge, Phys. Rev. 25, 240 (1925). Mecke, Zeits. f. Physik 32, 823 (1925).
g Since this paper has been sent to the editor, Professor Birge has kindly brought to the
writer's notice the fact that the equation r0'~0= C, where C has a diferent value for each
molecule, is a better fit for some data than Eq. (17). In other words, the slope of the curve
log r0 plotted against log cu0 is nearer two for each individual molecule, but the slope of the
band representing all molecules is nearer three, as given by Eq. (17). However, at least
one value of r0 must be known for a molecule before C can be known. Therefore Eq. (17)
presents the only means of obtaining a value for r0 for a molecule when no empirical value of
r0 for that molecule is available; and so it is useful, even if the probable error of the value so
obtained were rather larger than the above least squares analysis would indicate.
' Mulliken, Phys. Rev. 32, 206 (1928) and Birge, Int. Crit. Tables, Vol. V, 411, (1929).
62 PHILIP M. MORSE
TABLE I "
No. Mol. State A D
wave-nos. wave-nos,
rp obs. rp calc.
A units A units
refer-
ence
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
BeO
BeO
BO
BO
BO
Alo
Alo
C2
Cg
CN
CN
CN
CO
CO
CO
CO
CO
CO
CO+
CO+
CO+
Fg
F~
Hg
Hg
Hg
H2
Hg
Hg
Hg
Hg
Hg
H2
H2+
Ig
Ig
Ng
Ng
Ng
N2
N2
N +
N +
NO
NO
NO
NO
02
Og
Og
Q +
Q +
0 +
Q +
SiN
SiN
lg
lg
2g
2Q
2g
3II
3II
2g
2II;
2g
lg
3II
3Z?
~n
ly
~II
2g
2II.
2g
lg
1g
i'm
2~x
2'll
23K
(C)
33II
3~n
43@
5'll
6&n
lg
3gP
~n
3II
3II„
2g
2 g~
2II„
2+
2II„
3g
lg)
3+
2rr
~II
41600
72200
74200
59800
82500
33100
70000
55900
49100
75600
68700
78600
90500
99200
94900
96700
108800
145000
193100
177300
189100
33600
24800
38000
116900
112000
118200
119200
131500
225000
136500
140400
143000
143700
19100
20300
94300
103200
119100
126000
150300
202800
224500
61300
149200
80300
102100
53000
54500
59000
164400
165400
174800
187700
49400
38200
42300
51700
75100
36800
40000
33600
49800
56700
40600
76600
5520Q
53900
91600
51600
38300
32600
28800
46300
79200
42700
29600
34200
1420
40100
27600
19500
24600
21300
21200
114500
2040Q
21800
23000
20600
19200
4770
95500
37600
51000
51200
46100
67000
62900
62200
106200
35400
50000
53800
42100
10100
561QO
14300
23800
19700
50000
14500
2.12
1.76
2.14
2.04
1.99
2.06
1.51
2.14
2.65
2.32
2.30
2.88
2.29
2.45
1.93
2.67
4.55
2.86
2.91
2.42
3.17
2.39
3.21
1.85
.69
1.48
1.44
1.41
1.37
.61
1.38
1.32
1.28
1.36
1.50
1.80
2.56
2.42
2.41
2.46
3.31
2.62
3.10
2.55
2.42
1.82
3.49
2.34
2.39
2.44
2.82
2.57
2.31
2.93
1.92
3.15
1.33
1.36
1.21
1.36
1.31
1.62
1.66
1.31
1.27
1.17
1.15
1.15
1.24
1.12
1.11
1.25
1.17
1.4
.76
1.31
.97
1.08
1.06
1.14
1.14
1.17
1.17
1.06
2.66
3.01
1.21
1.15
1.12
1.08
1,15
1.07
1.42
1.21
1.23
1.61
1.57
1.58
1.27
1.30
1.17
1.34
1.33
1.46
1.52
1.23
1.19
1.14
1.20
1.12
1.12
1.20
1.37
1.26
1.12
1.16
1.11
1.25
1.21
1.37
2.11
.89
1.31
1.08
1.05
1.08
1.09
1.08
1.10
1.10
1.11
1.11
2.41
2.86
1.09
1.28
1.21
1.21
1.14
1.11
1.08
1.16
1.09
1.43
1.09
1.24
1.29
1.62
1,16
1.50
1.43
1.37
1.38
1.43
(14)
(14)
(9)
(9)
(9)
(15)
(»)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(11)
(16)
(11)
(11)
(11)
(11)
(11)
(11)
(11)
(11)
(11)
(13)
(13)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(9)
(12)
(12)
"It will be noticed that the molecules for which rp calc. differs markedly from rp obs. are
the least symmetric molecules in the list [ie., Mj M&/(M&+ M&) differs considerably from (M&+
M&)/4j. Professor R. S. Mulliken has kindly suggested to the writer that perhaps the rule
enunciated in Eq. (17) above only holds accurately for molecules where M& is about equal to
M&. Certainly the calculated values of rp for the perfectly symmetric molecules 02, H2 and N2
give the most consistent check with the experimental values. To apply the rule to very unsymmetric
molecules it may be necessary to introduce an "unsymmetry factor" of the type
f4M&M&/(M&+M2)']'~' into the term rp'cop. Curiously enough, those levels for which the above
rule is not satisfactory are the ones whose vibrational levels fit Eq. (13) least satisfactorily.
VIBRATIONAL LEVELS OIi DIATOMIC MOLZCULZS 63
Since six different neutral molecules and two different molecular ions
were used in the analysis, it would appear that Eis independent of molecular
weight, of the electronic state, and of the net charge on the molecule. An
independent check of this rule is discussed in the following paper.
RESULTS
A table is given from which the potential energy curves can be plotted
by means of the equation
jV —/+DE —2~(1—10) (18)
~0 is given in Angstrom units and D, coo and ~ox are given in wave-numbers.
The values of ro in the column marked ro calc. were calculated from coo by
means of rule (1'I). The striking agreement between these values and the
experimental values is to be noted. A is given so that all curves of neutral
molecule and ion are reckoned from the lowest vibration level of the lowest
electronic state of the neutral molecule. Figure 1 shows the levels for N2
and N2+ plotted from these data.
43
200000—
I
100000
40
39
36
37
00 i 2
Nuclear separation (Angstrom Units}
Fig. 1. Potential energy curves for nitrogen. Energy in wave-number and nuclear separation
in Angstrom units. Numbers on curves refer to Table I, ,
RoTATroNAL LEvELs
When the rotational quantum number p is different from zero, the potential
energy Ein Eq. (4) is increased by an amount E; =j(j+1)k'/8s.2pr02.
inasmuch as this increase only affects the wave function to an appreciable
"Birge, Proc. Nat. Acad. 14, 12 (1928).
» Jenkins and Laszlo, Proc. Roy. Soc. A122, 103 (1929).
"Birge, Molecular Spectra in Gases, N. R. C. Bull. SV, 230 (1927).
"Rosentl .l and Jenkins, Phys, Rev, 33, 163 (1929).
15 Pomeroy, Phys. Rev. 29, 57 (1927).
16 Hyman and Birge, Nature 123, 277 (1929).
PHILIP M. MORSE
extent in the region near r=rp (where W)E), it can be expanded about this
point,
fpj'(j +1) (r r,) —(r —r,)'
E2=—-- 1 —2 +3 r
Sm' pro' ro r2
In the range where E; has any appreciable effect, r —ro is small compared
to rp, and since I2'g(g+1)/8002 pr02, (which can be called R) is small compared
to E for the usual values of j, this expansion can be added to the expansion
for E, giving for the first two terms
E+Er= —D+R —R /Da rp +a (D —R)(r —rp —R/rpa D)
plus terms in higher powers of (r rp). —
These two terms can be considered as the first two terms of the expansion
of
E+E —(D R+R2/Da2r 2)&
—2a(r ra R/rpa D)——
2(D R+R2/Da2r 2)&
—a(r rp Rlrpa D)— —
(19)
indicating that to the first approximation D has decreased to D R+R'/rp'a—'D
and that r, has increased to r, +R/r, a'D The resu. ltant energy levels will be,
to the first approximation
I/2
W(nj ) = D+R+(a—h/220) (n+ ,')—p
a'h'(n+—')'/820'14 — R'/Da'—ro'
= —D+hppp(n+2) [1—h000(n+ 22)/4D hjp(j+1)/—16202DI2rp'J (20)
+( hjp(j +1) 8/ prppr 0) [1 hjp(j+1)/16—204I22rp'400'j.
This agrees with the general Kratzer" formula to the first approximation.
(i.e., as far as the above expansion is written).
The energy levels required by Eqs. (13) or (20) agree quite well with the
experimentally determined levels for most molecules up to quite high values
of n In ot. her words the AW(n) curve is a straight line for a considerable
distance as n is increased. This indicates that the potential energy is effectively
that given by Eq. (4) for a large range of r.
However there are some electronic states, usually the normal levels of
the molecules, whose 6W(n) curves are not straight lines, and therefore whose
potential energy curves deviate somewhat from the form given by Eq. (4).
This is not surprising, however, for the potential energies of initial states
have usually much deeper minima than the rest, and would be expected to
deviate most markedly from the standard form.
Such deviations from the straight line ~W curve can be considered as
due to an additional term in the potential energy of the form E'=8/r+
C/r'+, where B and C etc., are very small compared to D. This perturbation
can be dealt with in the same manner as E; has been dealt with,
and the values of 8, C, , can be found by comparing the resulting formula
for DW(n) with the data.
In most cases, however, the AW(n) curve deviates so little from the
straight line that such a calculation is not necessary.
"Kratzer, Zeits. f. Physik 3, 289 (1920).